We report the serendipitous detection of an H$_{2}$-bearing damped Lyman-$alpha$ absorber at z = 0.576 in the spectrum of the QSO J0111-0316 in the Cosmic Ultraviolet Baryon Survey. Spectroscopic observations from HST-COS in the far-ultraviolet reveal a damped absorber with log[N(HI)/cm^-2] = 20.1 +/- 0.2 and log[N(H$_{2}$)/cm^-2] = 18.97 (-0.06, +0.05). The diffuse molecular gas is found in two velocity components separated by dv = 60 km/s, with >99.9% of the total H$_{2}$ column density concentrated in one component. At a metallicity of $approx$ 50% of solar, there is evidence for Fe enhancement and dust depletion, with a dust-to-gas ratio $kappa_{text{O}} approx$ 0.4. A galaxy redshift survey conducted with IMACS and LDSS-3C on Magellan reveals an overdensity of nine galaxies at projected distance d <= 600 proper kpc (pkpc) and line-of-sight velocity offset dv$_{g}$ <= 300 km/s from the absorber. The closest is a massive, early-type galaxy at d = 41 pkpc which contains $approx$ 70% of the total stellar mass identified at d <= 310 pkpc of the H$_{2}$ absorber. The close proximity of the H$_{2}$-bearing gas to the quiescent galaxy and the Fe-enhanced chemical abundance pattern of the absorber suggest a physical connection, in contrast to a picture in which DLAs are primarily associated with gas-rich dwarfs. This case study illustrates that deep galaxy redshift surveys are needed to gain insight into the diverse environments that host dense and potentially star-forming gas.
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